Merging a shape onto another shape and removing excess polygons?
You can bool the shapes directly in Edit Mode, as long as they exist in one single mesh already. If you have two different meshes, joint them by selecting both and hitting Ctrl + J. Then make sure the normals are all pointing outwards (Ctrl + N recalculates normals if they are off), otherwise the boolean operation fails.

Now, select one of the meshes: Select one face, and using Ctrl + L select all connected faces), and using the Spacebar menu search for Boolean, like this: